A Solution to Yamakami's Problem on Advised Context-free Languages

TL;DR

This paper solves an open problem by showing the existence of a CFL-immune set in the intersection of two context-free languages that is not in CFL with advice, using the swapping lemma and nested palindromes.

Contribution

It provides an affirmative answer to Yamakami's problem on the complexity of CFL(2) relative to CFL/n, introducing new proof techniques.

Findings

Established the existence of a CFL-immune set in CFL(2) - CFL/n.

Demonstrated the effectiveness of the swapping lemma in context-free language analysis.

Showed the utility of nested palindromes in complexity proofs.

Abstract

Yamakami [2011, Theoret. Comput. Sci.] studies context-free languages with advice functions. Here, the length of an advice is assumed to be the same as that of an input. Let CFL and CFL/n denote the class of all context-free languages and that with advice functions, respectively. We let CFL(2) denote the class of intersections of two context-free languages. An interesting direction of a research is asking how complex CFL(2) is, relative to CFL. Yamakami raised a problem whether there is a CFL-immune set in CFL(2) - CFL/n. The best known so far is that LSPACE - CFL/n has a CFL-immune set, where LSPACE denotes the class of languages recognized in logarithmic-space. We present an affirmative solution to his problem. Two key concepts of our proof are the nested palindrome and Yamakami's swapping lemma. The swapping lemma is applicable to the setting where the pumping lemma (Bar-Hillel's lemma) does not work. Our proof is an example showing how useful the swapping lemma is.